ma gavte la nata (thing)

In standard Italian it would be "Ma levati il tappo" o "Ma cavati il turacciolo". Your average Italian would not understand the expression, because there is no word in Standard Italian akin to nata (on the other hand the relationship between gavte and cavati is quite obvious).
This being Umberto Eco's work, I would suspect the dialect to be from Alessandria. The people from Piedmont (Piemonte in Italian) are famous for their sarcastic wit.